Nonparametric estimation in null recurrent times series

We develop a nonparametric estimation theory in a non-stationary environment, more precisely in the framework of null recurrent Markov chains. An essential tool is the split chain, which makes it possible to decompose the times series under consideration in independent and identical parts. A tail condition on the distribution of the recurrence time is introduced. This condition makes it possible to prove weak convergence results for series of functions of the process depending on a smoothing parameter. These limit results are subsequently used to obtain consistency and asymptotic normality for local density estimators and for estimators of the conditional mean and the conditional variance. In contra-distinction to the parametric case, the convergence rate is slower than in the stationary case, and it is directly linked to the tail behaviour of the recurrence time.